Virtual Network Embedding Through Topology-AwareNode Ranking

Xiang Cheng, Sen Su∗, Zhongbao Zhang,Hanchi Wang, Fangchun Yang

Beijing University of Posts andTelecommunications

{chengxiang,susen, zhongbaozb,luigiking, fcyang}@bupt.edu.cn

Yan Luo, Jie WangUniversity of Massachusetts Lowell

yan luo@uml.edu, wang@cs.uml.edu

ABSTRACTVirtualizing and sharing networked resources have become agrowing trend that reshapes the computing and networkingarchitectures. Embedding multiple virtual networks (VNs)on a shared substrate is a challenging problem on cloud com-puting platforms and large-scale sliceable network testbeds.In this paper we apply the Markov Random Walk (RW)model to rank a network node based on its resource andtopological attributes. This novel topology-aware node rank-ing measure reflects the relative importance of the node. Us-ing node ranking we devise two VN embedding algorithms.The first algorithm maps virtual nodes to substrate nodesaccording to their ranks, then embeds the virtual links be-tween the mapped nodes by finding shortest paths with un-splittable paths and solving the multi-commodity flow prob-lem with splittable paths. The second algorithm is a back-tracking VN embedding algorithm based on breadth-firstsearch, which embeds the virtual nodes and links during thesame stage using node ranks. Extensive simulation experi-ments show that the topology-aware node rank is a betterresource measure and the proposed RW-based algorithms in-crease the long-term average revenue and acceptance ratiocompared to the existing embedding algorithms.

Categories and Subject DescriptorsC.2.5 [Computer-Communication Networks]: Local andWide-Area Networks; G.1.6 [Numerical Analysis]: Opti-mization

General TermsAlgorithms; Design; Performance

KeywordsNetwork Virtualization; Cloud Computing; Virtual NetworkEmbedding; Topology-aware; Random Walk; Markov Chain

1. INTRODUCTIONSharing virtualized resources enables new computing and

networking paradigms such as cloud based computing plat-forms [1] and sliceable network testbeds [15]. Users of acloud platform or a network infrastructure request theirshare of resources including CPU capacities, storage space,network bandwidth, etc., while the infrastructure providers

∗Corresponding author of this paper is Prof. Sen Su

make their best effort to serve the requests, which are alsoknown as virtual networks (VNs). The allocation of re-sources to VNs in such a virtualization environment is criti-cal to both users’ computation needs and the resource providers’monetary gain.

In the multi-tenant network virtualization environments,infrastructure providers (InPs) (e.g., cloud providers) andservice providers (SPs) (e.g., cloud users/tenants) play twodecoupled roles, namely, InPs manage the physical infras-tructure while SPs create VNs and offer end-to-end services[27, 14, 8]. Mapping VN requests of the SPs onto the sub-strate network of the InPs, also known as VN embedding,is NP-hard[28, 5]. Thus, devising heuristics has become themain line of research in VN embedding [13, 29, 22, 28]. Theearly algorithms measure the resource of a node by its CPUcapacity, or bandwidth, or both, without considering thetopological structure of the VNs and the underlying sub-strate network. Yet the topological attributes of nodes havesignificant impact on the success and efficiency of mappingoutcomes. It would make sense to measure a node’s re-sources and its topological attributes at the same time.

Inspired by PageRank used by Google’s search engine,which measures the popularity of web pages based on Markovrandom walks, we use the same theory to measure topology-aware resource ranking of a node, called NodeRank, whichreflects the resource and quality of connections of a node.PageRank considers a link from page A to page B as a vote,and a page is considered important if a number of importantpages vote to it. In such a way, the topology of the worldwide web influences the PageRank of a web page. In VN em-bedding, if a node links forward to a number of nodes withrelatively high importance, this node would also be consid-ered important, where the importance refers to the relativeresource quality of a node. We will take into account notonly the availability or requirements of the CPU and link re-sources of the node, but also its topological characteristics,i.e., the quality of its neighbors. Treating the connectivitybetween two nodes as a Markov chain transition with certainprobability, we can calculate the relative resource quality ofa node with a Markov chain model based on the topology ofthe network.

We devise two new VN embedding algorithms called RW-MaxMatch and RW-BFS based on NodeRanks. They firstcompute the node rank for each node in the VN requestand for each node in the residual substrate network. RW-MaxMatch is a two-stage VN embedding algorithm. In thefirst stage it maps a virtual node with the highest rank to

ACM SIGCOMM Computer Communication Review 39 Volume 41, Number 2, April 2011

a substrate node with the highest rank, a virtual node withthe second highest rank to a substrate node with the sec-ond highest rank, and continues in this manner for the restof the virtual nodes. In the second stage it embeds vir-tual links using the shortest path algorithm if path splitting[28] is not supported by the substrate network, or using themulti-commodity flow algorithm if the substrate supportspath splitting. Similar to the existing two-stage VNE al-gorithms, RW-MaxMatch may lead to higher substrate net-work resource consumption and restrict the ability of thesubstrate to accept additional future requests. RW-BFS canhelp solve this problem. It is a backtracking VN embeddingalgorithm based on breadth-first search that maps virtualnodes and virtual links during the same stage, aiming toincrease the resource utilization of the substrate resource.Extensive simulation experiments show that the topology-aware node rank is a better node resource measure and theproposed RW-based algorithms increase the long-term av-erage revenue and acceptance ratio compared to existingembedding algorithms.

This paper presents the following major contributions:

• We formulate a Markov random walk model to com-pute topology-aware resource ranking of nodes in anetwork, which serves as the basis of embedding vir-tual networks on substrate networks. To the best ofour knowledge, this work is the first to apply randomwalks in solving VNE problems.

• We devise two VNE algorithms based on topology-aware node ranks. Both two-stage and one-stage map-ping strategies are investigated.

• We conduct a thorough comparison between our algo-rithms and a wide range of existing algorithms throughextensive simulations. We design a VNE simulator andmake it publicly available to the research community.

The rest of the paper is organized as follows. In Section2, we discuss the related work. Section 3 presents the net-work model and formalizes the VN embedding problem. InSection 4 we present the method of computing the topology-aware resource ranks of nodes using the random walk model.Section 5 describes RW-MaxMatch and RW-BFS. The VNembedding algorithms are evaluated in Section 6. Section 7concludes the paper.

2. RELATED WORKThe VN embedding problem is similar to the virtual pri-

vate network (VPN) provisioning problem [18]. The ma-jor difference between them is on resource constraints. In atypical VPN request, the only resource constraints are band-width requirements from sources to destinations specified bya traffic matrix. There are typically no resource constraintson the nodes (e.g., CPU) and their locations. Another sim-ilar problem is the network testbed mapping problem. TheAssign algorithm [24] used in the Emulab testbed considersconstraints on both nodes and links, where the node con-straint is provided as the exclusive use of nodes, i.e., differ-ent virtual networks cannot share the same substrate node.VN embedding, however, allows substrate nodes and linksto be shared by multiple VNs.

Early studies on VN embedding either assume that theVN requests are known in advance (an offline version) [29,

22]; or deal with at most one type of constraints (node orlink) [13, 29, 22]; or perform no admission control when theresource of the substrate network is insufficient [13, 29, 22];or focus only on the backbone-star topology [22].

Without reducing the problem space, Yu et al. [28] intro-duce the mechanisms of substrate supporting path splittingand migration. Chowdhury et al. [9], while consideringthe same online VN embedding problem space as in [28],also consider location requirements of virtual nodes and usemixed integer programming (MIP) to solve the VN embed-ding problem.

Lischka et al. [21] model the topology of the substrateand the virtual network as a directed graph, and propose aVN embedding algorithm based on subgraph isomorphismwhich maps nodes and links during the same stage. Theiralgorithm can be seen as a extended version of the classic VFgraph matching algorithms [10], where link-on-link mappinghas been relaxed.

Houidi et al. [20] present a distributed VN embeddingalgorithm that achieves embedding through communicat-ing and exchanging messages between agent-based substratenodes. Although centralized algorithms could suffer from asingle point of failure, the performance and scalability of theproposed distributed algorithm compare unfavorably withthose of the centralized algorithms.

To maximize the aggregate performance across virtualnetworks, He et al. [19] propose an architectural frameworkcalled DaVinci to dynamically adapt virtual networks fora customized network substrate, where each substrate linkperiodically reassigns bandwidth among its virtual links.While on a smaller timescale, a distributed protocol is runin each VN to maximize the VN’s own performance objec-tive independently. DaVinci, however, does not have a nodeembedding stage.

Since the network condition change over time due to thearrival and departure of VNs, resources in the substrate net-work may become fragmented. Butt et al. [7] present atopology-aware measure using scaling factors for the sub-strate network, which identifies the bottleneck nodes andlinks in the substrate network. They then propose a setof algorithms for re-optimizing and re-embedding initially-rejected VN requests.

Recently, Guo et al. proposed a data center network vir-tualization architecture called SecondNet[17]. In Second-Net, the unit of resource allocation for multiple tenants inthe cloud is referred as virtual data center (VDC) whichconsists of virtual machines and virtual links. The VDC re-source allocation problem is close related to the VN embed-ding problem and the main difference is the problem scale.The VDC resource allocation algorithms proposed in [17]primarily focus on how to quickly allocate the resources tothe VDCs when a VDC has thousands of virtual machinesand the cloud infrastructure has tens to hundreds of thou-sands servers and switches, and how to satisfy the elasticityrequirement of VDCs. We believe that the topology-awarenode ranking method is general enough to be applied in theircontext to increase the possibility of satisfying the resourcerequirements of VDCs.

Page et al. [23] use random walks to rank the relativeimportance of web pages, where the rank of a page dependson the topological properties of the weighted links betweenthe pages, regardless of their content. A more general frame-work for this scheme was proposed in [11]. Another example

ACM SIGCOMM Computer Communication Review 40 Volume 41, Number 2, April 2011

a

b

c

20

20

15

12

15

VN Request 2

VN Request 1

A D

E

C

F

B

30

20

30

10

40

50

ba

c d

20

25

40

40

30 30

15

20

(c) (d)

A D

E

C

F

B

30

10

10

10

30

30c

b

a 20

25

25

25

20 20

15

20

Substrate Network

a b

dc

1010

20 20

15

10 10

15

(a) (b)

Substrate Network

Figure 1: Examples of VN embedding

of using random walks is to compute a set of topological sig-natures for each node in a graph, and it is also shown to beeffective for exact (and approximate) graph matching (see[16]). In a typical graph matching problem, there are eitherno weights on the nodes or links, or there are only visual fea-tures (e.g., RGB color space) contained in the nodes with-out links. Unlike PageRank and graph matching, however,in the VN embedding problem there are weights on bothnodes and links, and the weights are typically non-uniform.

Our work differs from the existing studies in three ways.First, we address the online VN embedding problem withadmission control, and do not need to reduce problem spaceas in [29, 22, 13]. Second, we consider both the resourceamount and topology properties of a node in a unified way torank the relative importance of a node, which will be lever-aged in the mapping procedure. Different from existing workthat only takes into consideration the resource (e.g., CPU,bandwidth, or both) of a node while neglecting its topologyproperty in computing the resource availability, our workmends this gap. Third, our topology-aware node rankingmeasure focuses on leveraging such a rank to benefit thecurrent VN embedding process rather than identifying thebottlenecks of the substrate nodes and links for the VN em-bedding reoptimization process proposed in [7]. In Section 4we will provide details how we apply the random walk modelto compute the topology-aware node resource ranks.

3. NETWORK MODEL AND PROBLEM DE-SCRIPTION

Substrate Networks. A substrate network can be repre-sented by a weighted undirected graph Gs = (Ns, Ls, A

ns , Al

s),where Ns is the set of substrate nodes and Ls the set of sub-strate links. The notations An

s and Als denote the attributes

of the substrate nodes and links, respectively. The attributesof the node include processing capacity, storage, and loca-tion. The typical attribute of the link is its bandwidth. Inthis paper we consider the available CPU capacity for thenode attribute and the available bandwidth for the link at-tribute as in most of the previous research. Denote by Ps

the set of all loop-free paths of the substrate network. Fig.1(b) presents a substrate network, where the numbers inrectangles are the available CPU resources at the nodes andthe numbers over the links represent available bandwidths.

Virtual Network Request. Similar to the substrate net-work, we use an undirected graph Gv = (Nv, Lv, Cn

v , Clv)

to denote a virtual network, where Nv is the set of virtualnodes and Lv the set of virtual links. Virtual nodes andlinks are associated with their capacity constraints, denotedby Cn

v and Clv, respectively. We also denote a VN request

by V NR(i)(Gv, ta, td), where ta is the arrival time of theVNR and td the duration of the VN staying in the sub-strate network. When the i-th VNR arrives, the substratenetwork should allocate resources to the VN to meet therequirements of the virtual nodes and links. If there areno sufficient substrate resources available, the VNR shouldbe rejected or postponed. The allocated substrate resourcesare released when the VN departs. There may be differentroles of all node in using VNs, such as directory or file serveretc. For simplicity, like most of the previous work [28, 9], weignore these dependencies. Fig. 1(a) and Fig. 1(c) presenttwo VN requests with node and link requirements.

VN Embedding Problem Description. The VN embed-ding problem is defined by a mapping M : Gv(Nv, Lv) →Gs(N

′s, P

′s) from Gv to a subset of Gs, where N

′s ⊂ Ns

and P′s ⊂ Ps. The mapping can be decomposed into two

mapping steps: (i) node mapping places the virtual nodesto different substrate nodes that satisfy the node resourceconstraints; and (ii) link mapping assigns the virtual linksto loop-free paths on the substrate that satisfy the link re-source requirements.

Fig. 1(a) and Fig. 1(b) show a VN embedding solutionfor VNR 1. Fig. 1(c) and Fig. 1(d) show another VNembedding solution for VNR 2, where residual resources arealso shown. Note that the virtual nodes of different VNRscan be mapped onto the same substrate node.

Objectives. The main objective of VN embedding is tomap the VNs to the substrate network to make efficient useof the substrate network resources, when the VN requestsarrive and depart over time.

Similar to the previous work in [29, 28, 9], the revenue ofaccepting a VNR at time t can be formulated by

R(Gv, t) =X

dv∈Nv

CPU(dv) +X

lv∈Lv

BW (Lv), (1)

where CPU(dv) and BW (Lv) are the CPU and the band-width requirements for virtual node dv and link lv, respec-tively.

The cost of accepting a VNR at time t is defined as thesum of the total substrate resources allocated to that VN:

C(Gv, t) =X

dv∈Nv

CPU(dv) +X

lv∈Lv

Xls∈Ls

BW (f lvls

, lv), (2)

where f lvls∈ {0, 1} and f lv

ls= 1 if substrate link ls allocated

bandwidth resource to virtual link lv, otherwise f lvls

= 0.

BW (f lvls

, lv) is the bandwidth allocated to lv from ls.From the InPs’ point of view, an efficient and effective on-

line VN embedding algorithm would maximize the revenueof InPs and increase the utilization of the substrate networkin the long run. Like the previous work in [28], the long-termaverage revenue is given by

limT→∞

PTt=0 R(Gv, t)

T. (3)

ACM SIGCOMM Computer Communication Review 41 Volume 41, Number 2, April 2011

n1

A1

D1 B1

C1

A2

D2 B2

C2

n2

(a) node n1 (b) node n2

Figure 2: Motivational example

The VNR acceptance ratio of the substrate network canbe defined by

limT→∞

PTt=0 V NRsPTt=0 V NR

, (4)

where V NRs is the number of VN requests successfully ac-cepted by the substrate network.

We also consider the long-term revenue to cost ratio toquantify the efficiency of resource utilization of the substratenetwork:

limT→∞

PTt=0 R(Gv, t)PTt=0 C(Gv, t)

. (5)

If the long-term average revenues of the VN embedding so-lutions are about the same, the higher VN acceptance ratioand R/C ratio are preferred.

4. TOPOLOGY-AWARE NODE RANKINGVN embedding incurs node mapping and link mapping.

Node mapping can be achieved by selecting substrate nodeswith sufficient CPU resources, and link mapping requiressufficient link resource on both of the selected nodes and thepath between any two selected substrate nodes. Most earlypublications (e.g., [28]) perform node mappings and linkmappings at two different stages, where nodes are selectedfirst at the node-mapping stage, and link allocation and pathselection are done at the link-mapping stage. We take a dif-ferent approach by incorporating topology attributes dur-ing the node mapping stage, aiming to improve the successrate and efficiency of link mapping. A motivational exam-ple is illustrated in Fig. 2, where larger nodes are nodeswith more CPU resources and the wider lines are links withmore bandwidth resources. Nodes n1 and n2 seem to havethe same resource availability if they are considered alone.However, n1 is a “better” node because the neighbors of n1,namely, A1, B1, and C1, have more resources than those ofn2’s neighbors, and so mapping a virtual node to n1 has ahigher chance to achieve a successful link mapping.

We define the notion of node rank to measure the resourceavailability of a node. Intuitively, the rank of a given nodeu is determined by its CPU power and its collective band-width of outgoing links. It is also affected by the ranks ofthe nodes that can be reached from u. We model the firstusing the product of its CPU and collective bandwidth ofoutgoing links as in [28]. We model the second by dividingreachable nodes into two groups, that is, the nodes that areincident to the outgoing links from u and the nodes that canbe reached from u via multiple hops. Modeling connectivityis challenging, and in this paper we define a jumping prob-ability to model the likelihood of a node that is reachable

from u via multiple hops and define a forward probabilityto model the influence of the neighboring nodes from u’sforward links. In particular, let

H(u) = CPU(u)X

l∈L(u)

BW (l), (6)

where, on a substrate network, L(u) is the set of all theoutgoing links of u, CPU(u) is the remaining CPU resourceof u, and BW (l) is the unoccupied bandwidth resource oflink l. On a virtual node, CPU(u) and BW (l) are the ca-pacity constraints of the node u, respectively. The initialNodeRank value for node u can be computed by

NR(0)(u) =H(u)P

v∈V

H(v). (7)

Let u, v ∈ V be two different nodes. Let

pJuv =

H(v)Pw∈V

H(w), (8)

pFuv =

H(v)Pw∈nbr1(u)

H(w), (9)

where pJuv denotes the jumping probability from node u to

land on node v, nbr1(u) = {v | (u, v) ∈ E}, and pFuv the

forward probability from node u to node v with (u, v) ∈ E.Clearly,

Xv∈V

pJuv = 1,

Xv∈nbr1(u)

pFuv = 1.

The probabilities pJuv and pF

uv may be viewed as resourcevoting for node u from, respectively, any node reachablefrom u and node u’s neighboring nodes. The voting froma non-neighboring node implies that there should exist amulti-hop path between u and v. Thus, the topology infor-mation of the two nodes is also embedded in the probabili-ties.

For any node v ∈ V , let

NR(t+1)(v) =Xu∈V

pJuv · pJ

u ·NR(t)(u) +

Xu∈nbr1(v)

pFuv · pF

u ·NR(t)(u), (10)

where pJu + pF

u = 1, pJu ≥ 0, pF

u ≥ 0, and t = 0, 1, · · · . ThepJ

u and pFu are bias factors, and we will typically want to set

pJu to 0.15 and pF

u to 0.85 (for details see Section 6).For a network of n nodes with V = {v1, v2, · · · , vn}, let

NR(t)i = NR(t)(vi) and denote the vector of node ranks at

iteration t by NR(t) = (NR(t)1 , NR

(t)2 , · · · , NR

(t)n )T , where

t = 0, 1, · · · . We have

NR(t+1) = T ·NR(t),

where T is a one-step transition matrix of the Markov chain

ACM SIGCOMM Computer Communication Review 42 Volume 41, Number 2, April 2011

defined by

T =

0BBB@

pJ11 pJ

12 · · · pJ1n

pJ21 pJ

22 · · · pJ2n

......

. . ....

pJn1 pJ

n2 · · · pJnn

1CCCA ·

0BBB@

pJ1 0 · · · 00 pJ

2 · · · 0...

.... . .

...0 0 · · · pJ

n

1CCCA +

0BBB@

0 pF12 · · · pF

1n

pF21 0 · · · pF

2n

......

. . ....

pFn1 pF

n2 · · · 0

1CCCA ·

0BBB@

pF1 0 · · · 00 pF

2 · · · 0...

.... . .

...0 0 · · · pF

n

1CCCA (11)

Note that T is stable since it is a stochastic matrix hav-ing a maximum eigenvalue equal to one. This guaranteesthat the above recurrence relation converges to NR(∗) =

(NR(∗)1 , NR

(∗)2 , · · · , NR

(∗)n )T , the steady state distribution

[25]. This can be computed using a classic iterative scheme[23], given by Algorithm 1.

Algorithm 1 The NodeRank Computing Method

1: Given a positive value ε, i← 02: repeat3: NR(i+1) ← T ·NR(i)

4: δ ← ‖NR(i+1) −NR(i)‖5: i + +6: until δ < ε

5. MAPPING WITH NODERANK:RW-MAXMATCH AND RW-BFS ALGO-RITHMS

Taking advantage of the topology-aware node resourcerank NR(∗), we devise two new VN embedding algorithmscalled RW-MaxMatch and RW-BFS, both taking the sub-strate network and a virtual network as input. Given beloware details of these two algorithms.

5.1 RW-MaxMatch

5.1.1 Description and DiscussionRW-MaxMatch is a two-stage VN embedding algorithm.

During the first phase, it first computes the NodeRank valuefor each virtual node and substrate node. It then sorts vir-tual nodes in non-increasing order according to their NodeR-ank values, and does the same on substrate nodes. Letm = |Nv| and n = |Ns|. Without loss of generality, letthe sorted virtual nodes be

N1v , N2

v , · · · , Nmv , where NR1 ≥ NR2 ≥ · · · ≥ NRm,

and let the sorted substrate nodes be

N1s , N2

s , · · · , Nns , where NR1 ≥ NR2 ≥ · · · ≥ NRn.

RW-MaxMatch maps N1v to N1

s , provided that the CPU ca-pacity and the link capacity of N1

s can meet what node N1v

asks for. It then maps N iv to N i

s in a similar way, wherei = 2, · · · , m. For convenience we call this mapping anL2S2 mapping (which stands for “large-to-large and small-to-small” mapping). This phase is executed in Algorithm2.

In the second phase, RW-MaxMatch maps virtual linksto substrate links (see Algorithm 3). The link embedding

stage is similar to the previous research [28, 26]. That is,find the just-fit shortest paths by searching the k-shortestpaths using binary search on values of k, until a path on thesubstrate network that satisfies the bandwidth requirementof the virtual link is found. If the substrate network sup-ports path splitting, we will use the multi-commodity flowalgorithm to embed the virtual links to the substrate linksbetween the mapped virtual nodes.

Algorithm 2 RW-MaxMatch Node Mapping Stage

1: Compute the NodeRank values of all nodes in both Gs

and Gv using Algorithm 1 with a given value of ε.2: Sort the nodes in Gs according to their NodeRank values

in non-increasing order.3: Sort the nodes in Gv according to their NodeRank values

in non-increasing order.4: Map virtual nodes to substrate nodes using the L2S2

mapping procedure.5: if node resource constraints satisfied then6: return NODE MAPPING SUCCESS7: else8: return NODE MAPPING FAILED9: end if

Algorithm 3 RW-MaxMatch Link Mapping Stage

1: if path unsplittable then2: Map virtual links using the k-shortest path algorithm.3: else4: Map virtual links using the multi-commodity flow al-

gorithm.5: end if6: if link resource constraints satisfied then7: return LINK MAPPING SUCCESS8: else9: return LINK MAPPING FAILED

10: end if

5.1.2 Time Complexity AnalysisNote that the iterative scheme (i.e., Algorithm 1) has been

shown to yield any desired precision ε with a number of itera-tions proportional to max{1,−logε} [6]. The k-shortest path[12] link mapping algorithm and the multi-commodity flowproblem [4] can both be solved in polynomial time. Thus,RW-MaxMatch is a polynomial-time algorithm in terms of|Gs|, |Gv|, and max{1,− log ε}.

5.2 RW-BFS

5.2.1 Description and DiscussionThe main framework of RW-BFS is presented in Algo-

rithm 4 and the details of the function called Match is pre-sented in Algorithm 5. Different from RW-MaxMatch, RW-BFS is a one-stage backtracking VN embedding algorithmbased on breadth-first search (BFS), which embeds virtualnodes and links at the same stage. The motivation of thisalgorithm is that a two-stage embedding algorithm maps allvirtual nodes first without considering the impact on the linkmapping stage, which may lead to unnecessary consumptionof bandwidth resources of the substrate network. For exam-ple, when the virtual nodes in the virtual network are very

ACM SIGCOMM Computer Communication Review 43 Volume 41, Number 2, April 2011

close to each other (e.g., just one-hop away), the distance oftwo mapped virtual nodes in the substrate network may beconsiderably long. Thus, the virtual link between the twonodes would have to be mapped to a long path, resulting ina waste of the bandwidth resources.

To help solve this problem, RW-BFS first computes theNodeRank values of the nodes in Gs and Gv. It then con-structs a breadth-first search tree of Gv, where the root nodeis the virtual node with the largest NodeRank value. At eachlevel of the search tree, nodes are sorted by their NodeRankvalues in non-increasing order. For each virtual node N i

v

in Gv, we build a candidate substrate node list consistingof the nodes whose available CPU capacity and the sum ofavailable bandwidth resources are at least as large as thoseof the virtual node. The substrate nodes in the list are alsosorted by their NodeRank values in non-increasing order. Inthe last step, we embed every virtual node in Gv to the sub-strate node that meets the CPU resource requirement andmaps the virtual links directly connected to a virtual nodeonto the substrate paths satisfying the bandwidth and con-nectivity constraints using the shortest path algorithm in abreadth-first search manner. If the virtual node is the rootnode of Gv, it is embedded to the substrate node with thelargest NodeRank value. If there is no suitable mapping forN i

v in its candidate substrate node list, we backtrack to theprevious virtual node N i−1

v , re-map it to another substratenode, and continue to map N i

v. Notice that the constantvalue Max Hop presented in the match function (Algorithm5) is the maximum distance from the mapped parent nodeof N i

v. The use of a hop bound is to reduce the search spaceof BFS. It can also avoid placing the virtual node too faraway from the already mapped nodes to save substrate linkresources.

Algorithm 4 RW-BFS

1: Compute the NodeRank values of all nodes in both Gs

and Gv using Algorithm 1 with a given value of ε.2: Construct the breadth-first searching tree of Gv.3: Sort all the nodes in Gv in non-increasing order in each

level of the breadth-first tree according to their NodeR-ank values.

4: backtrack count = 05: Select a positive integer Δ.6: for each node N i

v in Gv do7: Build the candidate substrate node list for N i

v

8: if Match(N iv)==1 then

9: Match(N i+1v )

10: else11: if backtrack count <= Δ then12: Match(N i−1

v )13: backtrack count++14: else15: return BFS FAILED16: end if17: end if18: end for19: return BFS SUCCESS

5.2.2 Time Complexity AnalysisIt is well known that backtracking algorithm has an ex-

ponential time complexity. To avoid exponential explosion,we introduce an upper bound θ to limit the number of noderemapping operations (in Algorithm 4 step 11).

Algorithm 5 The Details of Match Function

1: if N iv is the root then

2: map N iv onto the substrate node with the largest

NodeRank3: return MATCH SUCCESS4: end if5: k = 16: while k < Max Hop do7: for each N j

s which satisfies the k-hop constraint fromits mapped parent node in the candidate substratenode list of N i

v do8: if pair(N i

v, N js ) meets all the capacity constraints

then9: return MATCH SUCCESS

10: end if11: end for12: k++13: end while14: return MATCH FAILED

6. PERFORMANCE EVALUATIONIn this section we first describe the performance evalua-

tion environment, and then present our main evaluation re-sults. The experiments focus primarily on the performancecomparison of the proposed two algorithms with the existingones and the advantages of topology-aware node rank.

6.1 Evaluation SettingsWe implement two simulators to evaluate the performance

of our algorithms. The RW-MaxMatch simulator is a mod-ified version of the VN embedding simulator devised in [3],and the RW-BFS simulator is implemented from scratch.Both simulators are publicly available at [2].

The substrate network topology is configured to have 100nodes with about 500 links, corresponding to a medium sizedISP. We use the GT-ITM tool to generate substrate net-works. The CPU and bandwidth resources of the substratenodes and links are real numbers uniformly distributed be-tween 50 and 100. We assume that VNRs arrive following aPoisson process with an average arrival rate of 5 VNs per 100time units, and each one has an exponentially distributedlifetime with an average of 500 time units. In each VNR,the number of virtual nodes is determined by a uniform dis-tribution between 2 and 20. The average VN connectivity isfixed at 50%, which means that an n-node VN has n(n−1)/4links on average, similar with [28]. The CPU and bandwidthrequirements of virtual nodes and links are real numbers uni-formly distributed between 0 and 50. We run each of oursimulations for about 50000 time units, which correspondsto about 2500 VNRs on average in one instance of simula-tion.

Our simulation experiments evaluate eight algorithms listedin Table 1. Ten different instances are run for these nine al-gorithms and we record the arithmetic mean of the ten runsas the final result. Supporting location constraints is notour focus in this paper, thus we don’t compare our algo-rithms with algorithms proposed in [9], in which locationconstraints is taken into consideration. We have also con-sidered to make a comparison between our algorithms andthe virtual network embedding algorithm based on subgraphisomorphism proposed in [21]. However, the algorithm in[21] is limited to substrate networks and virtual networks

ACM SIGCOMM Computer Communication Review 44 Volume 41, Number 2, April 2011

Table 1: ComparisonsNotation Algorithm Description

RW-MM-SP

Use the RW model to compute NodeRankvalues. Two-stage mapping: first mapvirtual nodes with larger NodeRank val-ues to the substrate nodes with largerNodeRank values, then embed the virtuallinks using shortest path if the substratedoes not support path splitting. MM de-notes MaxMatch.

CB-MM-SPSimply use Equation (6) to compute noderesources rank. The other steps of CB-MM-SP is the same as RW-MM-SP. CBdenotes multiplying CPU by bandwidth.

RW-BFS

Use the RW model to compute NodeRankvalues. One-stage mapping: embed thevirtual nodes and links during the samestage based on breadth-first searching.

CB-BFSSimply use Equation (6) to compute noderesources rank. The other steps of CB-BFS is the same as RW-BFS.

RW-MM-MCF

Similar to RW-MM-SP, with path split-ting.

CB-MM-MCF Similar to CB-MM-SP, with path split-ting.

BL-SP The baseline algorithm proposed in [28].

BL-MCFThe baseline algorithm using MCF toembed virtual links.

modeled by directed graphs, thus a fair comparison is notapplicable and so we exclude it from the results.

The bias factors of pJv and pF

v presented in Equation (10)are used to balance between local and global node resources.Experiments show that setting pJ

v to 0.15 and pFv to 0.85,

the same values used in PageRank, will achieve the optimalresult. The value for the parameter ε presented in the it-erative scheme is set to 0.0001. Since the diameter of thesubstrate network is 3, we set the value of Max Hop in thematch function (Algorithm 5) to 3. To gain better mappingquality without consuming much execution time, the upperbound Δ, which limits the re-mapping steps in RW-BFS,is set to 3n, where n is the number of virtual nodes in theVNR.

6.2 Evaluation ResultsOur evaluation results quantify the efficiency of the two

algorithms we proposed (depicted in Fig. 3(a), Fig. 3(b),and Fig. 3(c)) and reveal the benefits of using the RW modelfor computing the node resource rank (depicted in Fig. 4(a),Fig. 4(b), and Fig. 4(c)). Several performance metrics forevaluation purposes are used, including the long-term aver-age revenue defined by Equation (3), the VNR acceptanceratio defined by Equation (4), the long term R/C ratio de-fined by Equation (5), and the runtime of these algorithms.We summarize the key observations from our simulation asfollows.

6.2.1 Our new algorithms lead to higher acceptanceratio and average revenue

Fig. 3(a) and Fig. 3(b) show that the proposed algorithmsRW-MM-SP and RW-BFS can produce higher revenue andacceptance ratio than those of the baseline algorithms, re-gardless the path splittability of substrate network. Thesetwo graphs show that topology-aware node ranking plays animportant role in VN embedding process. We note that al-though RW-BFS produces the highest acceptance ratio, thelong-term average revenue of this algorithm is lower than

that of RW-MM-MCF. The reason is that RW-BFS tendsto accept more small VNRs and reject a number of largeVNRs, thus the residual resources on the substrate networkwould have larger fragmentation.

6.2.2 RW-BFS produces the hightest long-term R/Cratio

The reason is that RW-BFS uses breadth-first search tomap virtual nodes, and avoids mapping virtual links onto along substrate path that may cause more resource consump-tion. Fig. 3(a) shows that, in the case of un-splittable paths,RW-BFS produces the highest revenue, which has close rela-tionship with its high long-term R/C ratio (depicted in Fig.3(c)). Lower VN embedding cost saves more room for thefuture VNRs.

6.2.3 NodeRank is a better node resource measureComputing NodeRank leads to higher revenue and accep-

tance ratio than simply using Equation (6) as the measureof node resource. With respect to the two-stage VN embed-ding algorithms, Fig. 4(a) and Fig. 4(b) show that RW-MM-SP and RW-MM-MCF offer larger revenue and betteracceptance ratio than CB-MM-SP and CB-MM-MCF. TheNodeRank also contributes to the increase of revenue andacceptance ratio for one-stage VN embedding algorithms.With NodeRank, the resource ranking is not only deter-mined by the node itself, but also influenced by its neigh-bors. A node with higher NodeRank value tends to have aset of good neighbors with higher NodeRank values, whichmay increase probability of meeting the constraints of theVN request and thus benefit the VN embedding process.However, shown in Fig. 4(c), NodeRank makes no signifi-cant contributions to increasing R/C ratio.

6.2.4 The RW-based algorithms consume compara-ble runtime with others

Fig. 5 depicts the average execution time of these VNembedding algorithms run on the same PC. The RW-basedalgorithms consume nearly the same time as the CB-basedalgorithms, even though the RM-based algorithms need tocompute node ranks, both in the substrate network and vir-tual network, for each VNR. The iterative process of com-puting the node rank converges to a reasonable tolerance inroughly 7 iterations for a substrate network with 100 nodesand about 500 links, which is deemed reasonable.

7. CONCLUSIONIn this paper, we compute the topology-aware node re-

source rank based on Markov Random Walks. Using theresource rank of a node as its resource measurement, wepropose two new algorithms called RW-MaxMatch and RW-BFS. Simulation results show that our algorithms outper-form the previous approaches in terms of the long-term aver-age revenue and the VNR acceptance ratio. We also demon-strate the benefit of applying the RW model to compute thenode resource rank for achieving a higher acceptance ratioand higher revenue.

In this work, we define the amount of node resources pre-sented in Equation (6) the same as the previous work in [28].

ACM SIGCOMM Computer Communication Review 45 Volume 41, Number 2, April 2011

22

24

26

28

30

32

4 8 12 16 20 24 28 32 36 40 44 48

Ave

rage

Rev

enue

Time (1000 Time Unit)

RW-MM-SPRW-MM-MCF

RW-BFSBL-SP

BL-MCF

(a) The long-term average revenue com-parison

0.7

0.75

0.8

0.85

0.9

0.95

4 8 12 16 20 24 28 32 36 40 44 48

Req

uest

Acc

epta

nce

Rat

io

Time (1000 Time Unit)

RW-MM-SPRW-MM-MCF

RW-BFSBL-SP

BL-MCF

(b) The VNR acceptance ratio over timecomparison

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

4 8 12 16 20 24 28 32 36 40 44 48

R/C

Rat

io

Time (1000 Time Unit)

RW-MM-SPRW-MM-MCF

RW-BFSBL-SP

BL-MCF

(c) The long-term R/C ratio (Rev-enue/Cost) comparison

Figure 3: Comparison between the RW-based and the existing algorithms

24

25

26

27

28

29

30

4 8 12 16 20 24 28 32 36 40 44 48

Ave

rage

Rev

enue

Time (1000 Time Unit)

RW-MM-SPRW-MM-MCF

RW-BFSCB-MM-SP

CB-MM-MCFCB-BFS

(a) The long-term average revenue com-parison

0.75

0.8

0.85

0.9

4 8 12 16 20 24 28 32 36 40 44 48

Req

uest

Acc

epta

nce

Rat

io

Time (1000 Time Unit)

RW-MM-SPRW-MM-MCF

RW-BFSCB-MM-SP

CB-MM-MCFCB-BFS

(b) The VNR acceptance ratio over timecomparison

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

4 8 12 16 20 24 28 32 36 40 44 48

R/C

Rat

io

Time (1000 Time Unit)

RW-MM-SPRW-MM-MCF

RW-BFSCB-MM-SP

CB-MM-MCFCB-BFS

(c) The long-term R/C ratio (Rev-enue/Cost) comparison

Figure 4: Comparison between the RW-based and CB-based algorithms

0

5

10

15

20

25

30

35

RW-MM-SP

CB-MM-SP

BL-SP RW-MM-MCF

CB-MM-MCF

BL-MCF RW-BFS CB-BFS

Tim

e (m

in)

Virtual Network Embedding Algorithms

Figure 5: The execution time comparison of embed-ding algorithms with 95% confidence interval

But we also find using other methods to mark the node re-sources value, such as α·CPU + (1 − α)·BW (0 < α < 1),appears to get better results in our preliminary experiments.The value of α can be changed dynamically to adapt to thesubstrate network resource environment when mapping anew virtual network request, and such study is left for fur-ther development. Secondly, the different value of bias ex-pressing the probability whether a node choosing to jumpto a random node or following an edge from this node willalso be tested in our future work. Finally, we plan to extendour work by considering the domain-specific substrate re-source availability (e.g., fat-tree type network topology) andrequirements from user requests (e.g., multi-dimensional re-source requirements on CPU, storage, etc.).

AcknowledgmentThis work is supported in part by the 973 project of China(2009CB320504), Funds for Creative Research Groups ofChina (60821001), NSF CNS-0709001, NSF/BBN GENI sub-award, NSF CCF-0830314 and NSF CNS-1018422.

8. REFERENCES[1] Amazonec2,http://aws.amazon.com/ec2/.

[2] VNE-RW Simulatorhttp://int.bupt.edu.cn/˜sensu/vne-rw.html.

[3] VNE Simulatorhttp://www.cs.princeton.edu/˜minlanyu/embed.tar.gz.

[4] R.K. Ahuja, T.L. Magnanti, J.B. Orlin, and K. Weihe. Networkflows: theory, algorithms, and applications. Prentice hallEnglewood Cliffs, NJ, 1993.

[5] David G. Andersen. Theoretical approaches to nodeassignment. Unpublished Manuscript, December 2002.

[6] M. Bianchini, M. Gori, and F. Scarselli. Inside pagerank. ACMTransactions on Internet Technology (TOIT), 5(1):92–128,2005.

[7] N. Butt, M. Chowdhury, and R. Boutaba. Topology-Awarenessand Re-optimization Mechanism for Virtual NetworkEmbedding. In Networking 2010: 9th International Ifip Tc 6Networking Conference, Chennai, India, May 11-15, 2010,Proceedings, page 27. Not Avail, 2010.

[8] N.M.M.K. Chowdhury and R. Boutaba. Network virtualization:state of the art and research challenges. IEEECommunications magazine, 47(7):20–26, 2009.

[9] N.M.M.K. Chowdhury, M.R. Rahman, and R. Boutaba. Virtualnetwork embedding with coordinated node and link mapping.In IEEE INFOCOM, 2009.

[10] L.P. Cordella, P. Foggia, C. Sansone, and M. Vento. Animproved algorithm for matching large graphs. In 3rdIAPR-TC15 Workshop on Graph-based Representations inPattern Recognition, pages 149–159, 2001.

[11] M. Diligenti, M. Gori, M. Maggini, andD. di Ingegneria dell’Informazione. A unified probabilistic

ACM SIGCOMM Computer Communication Review 46 Volume 41, Number 2, April 2011

framework for web page scoring systems. IEEE Transactionson knowledge and data engineering, 16(1):4–16, 2004.

[12] D. Eppstein. Finding the k shortest paths. In Proc. of IEEESymposium on Foundations of Computer Science. IEEEComput. Soc. Press, 1994.

[13] J. Fan and M. Ammar. Dynamic topology configuration inservice overlay networks: A study of reconfiguration policies. InProc. IEEE INFOCOM, 2006.

[14] N. Feamster, L. Gao, and J. Rexford. How to lease the Internetin your spare time. ACM SIGCOMM ComputerCommunication Review, 37(1):64, 2007.

[15] Global Environment for Network Innovations. National ScienceFoundation, http://www.geni.net/, August 2005.

[16] M. Gori, M. Maggini, and L. Sarti. Exact and approximategraph matching using random walks. IEEE Transactions onPattern Analysis and Machine Intelligence, 27(7):1100–1111,2005.

[17] C. Guo, G. Lu, H.J. Wang, S. Yang, C. Kong, P. Sun, W. Wu,Y. Zhang, MSR Asia, and MSR Redmond. SecondNet: A DataCenter Network Virtualization Architecture with BandwidthGuarantees.

[18] A. Gupta, J. Kleinberg, A. Kumar, R. Rastogi, and B. Yener.Provisioning a virtual private network: A network designproblem for multicommodity flow. In Proceedings of thethirty-third annual ACM symposium on Theory of computing,pages 389–398. ACM New York, NY, USA, 2001.

[19] J. He, R. Zhang-Shen, Y. Li, C.Y. Lee, J. Rexford, andM. Chiang. Davinci: Dynamically adaptive virtual networks fora customized internet. In Proceedings of the 2008 ACMCoNEXT Conference, pages 1–12. ACM, 2008.

[20] I. Houidi, W. Louati, and D. Zeghlache. A distributed virtualnetwork mapping algorithm. In Proceedings of IEEE ICC,pages 5634–5640, 2008.

[21] J. Lischka and H. Karl. A virtual network mapping algorithmbased on subgraph isomorphism detection. In Proceedings ofthe 1st ACM workshop on Virtualized infrastructure systemsand architectures, pages 81–88. ACM, 2009.

[22] J. Lu and J. Turner. Efficient mapping of virtual networks ontoa shared substrate. Department of Computer Science andEngineering, Washington University in St. Louis, TechnicalReport WUCSE-2006, 35, 2006.

[23] L. Page, S. Brin, R. Motwani, and T. Winograd. The pagerankcitation ranking: Bringing order to the web. Technical report,Stanford Digital Library Technologies Project, 1998.

[24] R. Ricci, C. Alfeld, and J. Lepreau. A solver for the networktestbed mapping problem. ACM SIGCOMM ComputerCommunication Review, 33(2):81, 2003.

[25] E. Seneta. Non-negative matrices and Markov chains.Springer Verlag, 2006.

[26] W. Szeto, Y. Iraqi, and R. Boutaba. A multi-commodity flowbased approach to virtual network resource allocation. In Proc.GLOBECOM: IEEE Global Telecommunications Conference,2003.

[27] JS Turner and DE Taylor. Diversifying the internet. In IEEEGlobal Telecommunications Conference, 2005.GLOBECOM’05, volume 2.

[28] M. Yu, Y. Yi, J. Rexford, M. Chiang, et al. Rethinking virtualnetwork embedding: Substrate support for path splitting andmigration. COMPUTER COMMUNICATION REVIEW,38(2):17, 2008.

[29] Y. Zhu and M. Ammar. Algorithms for assigning substratenetwork resources to virtual network components. In Proc.IEEE INFOCOM, 2006.

ACM SIGCOMM Computer Communication Review 47 Volume 41, Number 2, April 2011